#7622904

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622904
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×Q₈	C₂⁵
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

3625 conjugacy classes of subgroups; only the 568 normal subgroups are shown.

normal subgroups only

This group has 3625 conjugacy classes of subgroups — too many to draw. Only its 568 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 64 points:

(1,2)(3,13)(4,18)(5,22)(6,27)(7,12)(8,16)(9,21)(10,26)(11,30)(14,43)(15,36)(17,50)(19,39)(20,55)(23,32)(24,54)(25,37)(28,34)(29,40)(31,59)(33,52)(35,56)(38,49)(41,46)(42,47)(44,63)(45,53)(48,61)(51,62)(57,58)(60,64);(1,3)(2,8)(4,14)(5,23)(6,17)(7,15)(9,31)(10,36)(11,38)(12,32)(13,22)(16,26)(18,53)(19,44)(20,48)(21,46)(24,45)(25,52)(27,33)(28,51)(29,49)(30,47)(34,60)(35,62)(37,42)(39,61)(40,50)(41,58)(43,55)(54,63)(56,59)(57,64);(1,4)(2,9)(3,14)(5,24)(6,28)(7,19)(8,31)(10,35)(11,39)(12,34)(13,41)(15,44)(16,48)(17,51)(18,27)(20,26)(21,30)(22,58)(23,45)(25,56)(29,57)(32,60)(33,53)(36,62)(37,54)(38,61)(40,55)(42,63)(43,50)(46,47)(49,64)(52,59);(4,19)(9,34)(14,44)(18,39)(20,24)(21,28)(31,60)(35,58)(41,62)(43,63)(45,48)(46,51)(53,61)(54,55)(56,57)(59,64);(3,15)(8,32)(13,36)(14,44)(16,23)(17,47)(31,60)(33,38)(41,62)(42,50)(43,63)(45,48)(46,51)(49,52)(53,61)(59,64)

Polycyclic presentation:

2^9|f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,e^b=ei,f^b=fi,g^b=ghi,d^c=di,f^c=fh,g^c=gi,g^f=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210828, #72227409, #72267976
maximal normal subgroup of	#72210828, #72227409, #72267976
maximal quotient of	#72210404, #72212257, #72213168, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530100);